Showing papers on "Compressibility published in 1973"

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

Abstract: A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972). Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions). The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for. At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants. A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included.

Compressibility of water as a function of temperature and pressure

Abstract: The isothermal compressibility of water from 0 to 100 °C and 0 to 1000 bar has been determined from Wilson's sound velocity measurements which have been normalized to Kell's 1 atm values. The isothermal compressibilities determined from the sound velocities have been fit, with a maximum deviation in compressibility of ± 0.016 × 10−6 bar−1, to an extended bulk modulus equation V 0 P/(V 0‐V P ) = B + A 1 P + A 2 P 2, where V 0 and V P are the specific volume at an applied pressure of zero and P; and B, A 1, and A 2 are temperature dependent constants. Our specific volume results are in reasonable agreement with the work of Kell and Whalley at low pressures; however, our results at high pressures (1000 bar) disagree by as much as 169 ppm (the average deviation is approximately 115 ppm). A comparison of the compressibilities indicates a parabolic shift in Kell and Whalley's work with a maximum of approximately 0.205 × 10−6 bar−1 at 400 bar and 5 °C. Since the velocity of sound data is extremely reliable (± 0.2 m/sec) and the maximum error in the compressibilities derived from the sound data is within ± 0.016 × 10−6 bar−1, our PVT results based upon the sound data are more accurate than any direct measurements made to date.

Flow of compressible fluid in porous elastic media

Abstract: A finite element formulation for flow of fluid in a porous elastic media has been derived from a Gurtin type variational principle. Biot's field equations for porous media have been used in which the constitutive relations include the compressibility of the fluid. It has been shown that by proper choice of the form of the compressibility of the fluid as a function of the state variables, this option of the formulation can be used to treat the partially saturated soil. The method is general with respect to geometry, boundary conditions and material properties. Finally, the results of a series of examples have been presented and compared with exact results to demonstrate accuracy and applicability.

Simple Compressibility Relation for Solids

Abstract: Isothermal compression data derived from shock-wave and static-compression measurements on metals exhibit a nearly precise linear relation between the logarithm of the bulk modulus and the specific volume up to volume changes of 40% As a result, solid isotherms can be accurately fitted or extrapolated in this range by means of two parameter functions of either a Birch or a modified Tait form

Fully developed viscous flow in coiled circular pipes

Abstract: The Navier-Stokes equations in stream-function/vorticity form were solved numerically by over-relaxation for the case of steady state, fully developed, isothermal, incompressible viscous Newtonian flow within a rigorously treated toroidal geometry. Solutions were obtained for curvature ratios ranging from 5 to 100 and for Dean numbers as low as I and as high as 1000. The Dean number was demonstrated to be the principal parameter to characterize toroidal flow; however, a second-order dependence upon the curvature ratio above that expressed in the Dean number was observed. Comparisons of the numerically computed axial-velocity profiles were made with experimental data. The cross-sectional pressure distribution was calculated, and a correlation is presented for a diametral pressure drop in terms of the Dean number.

The influence of an undrained change in stress on the pore pressure in porous media of low compressibility

Abstract: The author describes some earlier work directed at predicting and measuring the pore pressures that are set up by changes in the state of stress of soils, under undrained conditions. He shows how he has derived from first principles an expression of pore porosity ratio in terms of porosity, compressibility of the pore water, compressibility of the soil grains and compressibility of the soil skeleton. It is indicated how the same expression can be derived from Geertsma's equation relating the change in volume of pore space to the changes in total stress and pore pressure. It is stated that the methods of analysis used by Gassman led to conclusions that were consistent with the author's and Geertsma's findings. This is followed by a discussion of the principle of effective stress and, in particular, the importance of using the excess of total stress over pore pressure in the calculations rather than the effective stress relating to volume change. The author then discusses the predicted range of values of pore pressure ratio and concludes with a derivation of the pore pressure parameter as defined by Skempton.

A Computer Program for Earthquake Analysis of Gravity Dams Including Hydrodynamic Interaction

Abstract: : A computer program for earthquake analysis of gravity dams including effects of hydrodynamic interaction is presented. The dam-water system is idealized as two-dimensional in geometry, the material behavior is assumed to be linearly elastic, and the transverse horizontal as well as the vertical component of ground motion are considered. The dam is represented as a finite element system, whereas water in the reservoir is treated as a continum of infinite length in the upstream direction governed by the wave equation. Compressibility of water is considered resulting in governing equations for the dam depending on the excitation frequency. The analysis is performed in the frequency domain, first obtaining the frequency responses and then Fourier Synthesizing them by Fast Fourier Transform procedures to obtain responses to arbitrary ground motion. A listing of the computer program is included and the usage and capabilities are illustrated by examples. (Author)

Earthquake analysis of gravity dams including hydrodynamic interaction

Abstract: A general procedure for analysis of the response of gravity dams, including hydrodynamic interaction and compressibility of water, to the transverse horizontal and vertical components of earthquake ground motion is presented. The problem is reduced to one in two dimensions considering the transverse vibration of a monolith of a dam, and the material behaviour is assumed to be linearly elastic The complete system is considered as composed of two substructures—the dam, represented as a finite element system, and the reservoir, as a continuum of infinite length in the upstream direction governed by the wave equation. The structural displacements of the dam (including effects of water) are expressed as a linear combination of the modes of vibration of the dam with the reservoir empty. The effectiveness of this analytical formulation lies in its being able to produce excellent results by considering only the first few modes. The complex frequency response for the modal displacements are obtained first. The responses to arbitrary ground motion are subsequently obtained with the aid of the Fast Fourier Transform algorithm An example analysis is presented to illustrate results obtained from this method. It is concluded that the method is very effective and efficient and is capable of producing results to any desired degree of accuracy by including the necessary number of modes of vibration of the dam.

Compressibility of fifteen minerals to 45 kilobars

Abstract: We report here new measurements, by static methods, of the compressibility of 15 minerals to 45 kb. Data have been reported by P. W. Bridgman on a number of these minerals, but data on nine of them are reported here for the first time. Abnormally high compressibility, presumably owing to voids in the minerals, was found at low pressures. It is difficult to make corrections for this effect; therefore the accuracy of the data is not high.

Nonlinear streaming effects associated with oscillating cylinders

Abstract: This paper deals with nonlinear streaming effects associated with oscillatory motion in a viscous fluid. A previous theory by Holtsmark et al. (1954) for the streaming near a circular cylinder in an incompressible fluid of infinite extent is reconsidered and used to obtain new numerical results, which are compared with earlier observations. The regime of validity of this theory is considered. The condition to be satisfied by the Reynolds number is found to be less stringent than was previously supposed.The more recent theory by Wang (1968) based on the outer–inner expansion technique is discussed and corrected with the Stokes drift.The case of an incompressible fluid enclosed between two coaxial cylinders, one of which is oscillating, is considered in detail. New theoretical and experimental results are given for various values of the parameters involved (Reynolds number, amplitude and cylinder radii).

The Pressure and Deformation Profiles Between Two Normally Approaching Lubricated Cylinders

Abstract: The pressure and deformation profiles between two colliding lubricated cylinders are obtained by solving the coupled, time-dependent elastohydrodynamic equations with an iterative procedure. The analysis includes several effects which were not considered in a previous solution by Christensen [4], namely, the effect of surface velocities due to local deformation, the effect of the lubricant compressibility, and the effect of a lubricant with composite pressure-viscosity coefficients. It is found that the local approach velocity plays an important role during final stages of normal approach. It causes the lubricant to be entrapped within the contact region, and both the pressure and deformation profiles appear to converge to the Hertzian profile. The use of a smaller pressure-viscosity coefficient at high pressures reduces the sharp pressure gradient at the center of the contact, and produces a much milder variation of load with respect to the film thickness. The effect of compressibility of the lubricant is found to be relatively small.

Pressure-volume-temperature properties of amorphous polymers: empirical and theoretical predictions

Abstract: Equation of state data on several amorphous polymers, viz., polystyrene, poly(methyl methacrylate) and poly(vinyl chloride) are considered from two points of view. First in terms of the empiricalTait equation applied to the liquid as well as the glassy state with a single disposable and temperature dependent parameter. These results supersede and extend an earlier analysis. Good agreement between independent investigators is obtained for polystyrene. Secondly, the hole theory ofSimha- Somcynsky is applied to the liquid state of these polymers. The isobar at atmospheric pressure and the compressibility factorpV/T up to pressures of about 2 kbar are analyzed in detail and the numerical values of the characteristic volume (V*), temperature (T*), and pressure (p*) parameters are established. The evaluation of the theoretical expressions requires the numerical solution of a transcendental equation for the hole fraction as a function of the reduced volume and temperature, which arises from the extremum condition on the partition function. At atmospheric pressure, this computation is avoided and the evaluation facilitated by means of a simple and accurate interpolation formula for the theoretical volume-temperature relation. In this manner,V* andT* are readily obtained. A series of theoretical reduced isochores are presented in graphical form, which encompass a change in density of 15% and a temperature range of at least 200 degrees. A single pressure datum then provides an estimate ofp* and thus of the complete equation of state. Finally we note that the reducedTait parameterB, as computed from the theory, exhibits the exponential variation with temperature, observed experimentally.

On the atomization of liquids by high-velocity gases

Abstract: A mathematical model of the gas atomization of a liquid is developed. This model, in which the liquid is allowed to have infinite depth, takes into account both the compressibility of the gas and the viscosity of the liquid, enabling a prediction to be made of the expected drop size in the spray formed when a gas flows over a liquid. It gives good results for the air atomization of water and an order-of-magnitude agreement for that of molten metals. A universal curve is constructed for molten metals such that the predicted size of the most frequently occurring drop size can be found by evaluating a single formula. The effect of allowing the liquid to have finite depth is investigated in two cases of interest.

Equation for one-dimensional vertical flow of groundwater: 1. The Rigorous Theory

Abstract: A new mathematical derivation of the one-dimensional flow equation in an elastic, saturated, porous medium is presented. The approach involving the consideration of a fixed elemental volume in fixed coordinates is developed by starting from both Lagrangian and Eulerian definitions of the position vector. The confusion existing about these two fundamental points of view has so far led to theoretically erroneous results. The Lagrangian and Eulerian formulations prove to be equivalent and provide the same outcome if they are correctly interpreted and consistently applied. The rigorous equation, compared with Cooper's (1966) equation, turns out to contain an additional nonlinearity resulting from the correct expansion of the partial spatial derivative of the grain velocity. It is also shown that an approach based on a deforming element in fixed coordinates is simple and straightforward, since it does not introduce the grain velocity into the development. However, it needs a particular definition for the compressibility different from the classical one. It is proved that these two compressibilities are not equal; their mathematical link is derived.

Flow measuring method and apparatus

Abstract: Selected variables of a fluid flowing through a conduit are measured by transmitting acoustic pulses through the conduit wall and fluid stream along normal and oblique transmission paths between acoustic transducers located externally of the conduit, measuring the transit times of the pulses between the transducers, and combining these transit times with certain conduit and transducer parameters according to predetermined mathematical relationships to obtain the values of the variables The variables which may be measured are flow velocity, mass flow rate, sonic velocity of the fluid, fluid compressibility, fluid temperature, and fluid density

Aerodynamic sound and the low‐wavenumber wall‐pressure spectrum of nearly incompressible boundary‐layer turbulence

Abstract: The low‐wavenumber structure of fixed‐frequency cross‐spectral density of turbulent pressure fluctuations at a rigid plane boundary is determined for flows with small but finite Mach number. The method of matched asymptotic expansions is applied in the coordinate normal to the boundary. The boundary layer is an inner region, and prescribes an “effective” boundary velocity distribution for the outer region, which is governed by the acoustic wave equation. Those components of effective velocity with supersonic phase speeds account for the radiation of sound. For the inviscid infinite plate model, the wall‐pressure spectrum has a nonintegrable singularity at the acoustic critical wavenumber. Because of undamped contributions to point pressure from distant acoustic sources, in fact, the infinite model fails in an inviscid medium with any degree of compressibility. A large but finite model is considered, and the nonintegrable singularity at the critical wavenumber is removed. The spectrum coincides otherwise with the infinite plate result. The finite extent of the model can represent either a real geometrical limitation or the effect of damping over long distances. The intensity in the radiated field is shown to vary with the eighth power of velocity, but with a coefficient proportional to the logarithm of the characteristic in‐plane dimension.

Methods and apparatus for determining characteristics of subsurface earth formations

Abstract: In accordance with illustrative embodiments of the present invention, measurements of a plurality of earth formation parameters are combined to enable interpretation of the mechanical properties of the formations surrounding a borehole. Information concerning the mechanical properties of the formation may then be used for identifying mechanically competent formations. More particularly, sonic and density measurements are obtained and combined in a manner to provide data indicative of certain formation elastic moduli. These moduli, such as shear modulus and bulk compressibility of the formation provide an indication of the formation strength.

Elasticity of some mantle crystal structures. II.

Abstract: The single-crystal elastic constants are determined as a function of pressure and temperature for rutile structure germanium dioxide (GeO2). The data are qualitatively similar to those of rutile TiO2 measured by Manghnani (1969). The compressibility in the c direction is less than one-half that in the a direction, the pressure derivative of the shear constant is negative, and the pressure derivative of the bulk modulus has a relatively high value of about 6.2. According to an elastic strain energy theory, the negative shear modulus derivative implies that the kinetic barrier to diffusion decreases with increasing pressure.

Spatial structure and thermodynamic properties of a classical fluid of hard spheres with attractive square wells

Abstract: The spatial structure and the thermodynamic properties of a classical fluid of hard spheres with attractive square wells are analyzed, based on the Percus‐Yevick equation reformulated by Baxter. The equation of state, the isothermal compressibility, the internal energy, the direct correlation function, the radial distribution function, and the structure factor are evaluated over a wide range of temperatures and densities when the width of the attractive square well is half the diameter of the hard core. The temperature‐ and the density‐dependent behaviors of the obtained quantities are discussed. The equation of state is compared with theoretical results obtained by other authors. The pressures calculated via the compressibility equation are in satisfactory agreement with the pressures obtained from computer experiments both at the supercritical temperature kT/e=3.333 and at the subcritical temperature kT/e=1.17. Attempts are made to fit the pressures calculated via the compressibility equation to the exp...

Equation for one-dimensional vertical flow of groundwater: 2. Validity range of the diffusion equation

Abstract: The reliability in the operative field of the parabolic diffusion equation is investigated by solving the rigorous one-dimensional equation of groundwater flow in deforming soils. The dependence of the hydraulic conductivity on the specific weight of water is included. The grain velocity is correctly expanded first. This expansion leads to a nonlinear integrodifferential term. An iterative finite element technique of solution is then developed. The true time-dependent pressure head is compared to the standard one. The entire range of realistic variations for the formation parameters is carefully explored. It turns out that the usual equation gives satisfactory results in the vast majority of applications. The conditions underlying the approximated theory become critical only when the flow field is to be determined in highly compressible units for strong boundary pressure variations. In this case the solid material movements can no longer be considered small. The pressure head changes are faster than it would appear from the standard solution, and the consolidation process is more rapid than that in the classical Terzaghi's theory.

Steady waves in ductile porous solids

Abstract: A porous material is idealized as a suspension of voids in an incompressible ductile matrix. A constitutive equation for this ideal material is based on a rate‐dependent pore‐collapse relation obtained previously from a spherical model calculation. This theory is used to study the propagation of steady waves; closed‐form expressions are obtained for the Hugoniots and the differential equation for the steady wave profiles is integrated numerically. A feature of the theory is prediction of a finite compaction pressure Pc such that shocking to pressure P* causes partial compaction if P* < Pc and total compaction if P* ≥ Pc. We also discuss a qualitative difference in the behavior of the rate‐dependent energy for these two situations.

Developing turbulent flow in smooth pipes

Abstract: A numerical and experimental investigation of steady incompressible developing turbulent flow in smooth pipes is presented. Finite difference techniques are used to solve simultaneously the vorticity transport and stream function equations utilising a modified form of the Van Driest effective viscosity model. The numerical solutions are verified experimentally using air as a working fluid at pipe Reynolds 1 × 105, 2 × 105 and 3 × 105.